On the spectral Stefan-Florin problem with classical boundary condition. (English. Ukrainian original) Zbl 1291.35449
J. Math. Sci., New York 192, No. 4, 474-484 (2013); translation from Ukr. Mat. Visn. 10, No. 1, 129-143 (2013).
Summary: The purpose of this work is to study the spectral properties of the problem of transmission arising after the linearization of two-phase problems of Stefan and Florin with classical boundary condition on a small time interval. With the help of the operator methods of mathematical physics, a boundary-value problem is reduced to the study of the spectrum of a weakly perturbed compact self-adjoint operator in a Hilbert space. On the basis of the theorems of M. V. Keldysh and V. B. Lidskii, we have established the basis property of the system of eigen- and associated elements by Abel-Lidskii in some Hilbert space. It is proved that the spectrum is discrete with the single limiting point at infinity. It is situated on the positive semiaxis or, except for a finite number of eigenvalues, in the aperture angle \(\varepsilon\). The growth of the moduli of eigenvalues is estimated, and some asymptotic formulas are obtained.
Keywords:
spectral transmission problem; Green’s formula; compact operator; discrete spectrum; Hilbert space; Abel-Lidskii basisReferences:
| [1] | B. V. Bazalii and S. P. Degtyarev, ”On the Stefan problem with kinetic and classical conditions on the free boundary,” Ukr. Mat. Zh., 44, No. 2, 155–166 (1992). · Zbl 0755.35155 · doi:10.1007/BF01061737 |
| [2] | Yu. M. Berezansky, Expansions in Eigenfunctions of Self-Adjoint Operators, Amer. Math. Soc., Providence, RI, 1969. |
| [3] | M. Sh. Birman and M. Z. Solomyak, Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory, Amer. Math. Soc., Providence, RI, 1980. · Zbl 0426.46020 |
| [4] | M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Reidel, Dordrecht, 1987. · Zbl 0744.47017 |
| [5] | G. I. Bizhanova, ”On the classical solvability of one-dimensional Florin, Masket–Verigin, and Stefan problems with free boundary,” Zap. Nauch. Sem. POMI, No. 243, 30–60 (1997). · Zbl 0905.35098 |
| [6] | V. A. Florin, ”The densification of a soil madium and the filtration at a variable porosity with regard for the effect of bound water,” Izv. AN SSSR, OTN, No. 1, 1635–1649 (1951). |
| [7] | Functional Analysis [in Russian], edited by S. G. Krein, Nauka, Moscow, 1972. |
| [8] | I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Nonself-Adjoint Operators, Amer. Math. Soc., Providence, RI, 1969. |
| [9] | V. B. Lidskii, ”On the summability of series in principal vectors of nonself-adjoint operators,” Trudy Mosk. Mat. Obshch., 11, 3–35 (1962). |
| [10] | N. D. Kopachevsky, S. G. Krein, and Ngo Zui Kan, Operator Methods in Linear Hydrodynamics. Evolutionary and Spectral problems [in Russian], Nauka, Moscow, 1989. |
| [11] | N. D. Kopachevsky and V. I. Voytitsky, ”On the modified spectral Stefan problem and its abstract generalizations,” in Operator Theory: Advances and Applications, Vol. 191, Birkhäuser, Basel, 2009, pp. 373–386. · Zbl 1179.35202 |
| [12] | E. V. Radkevich, ”On the conditions of existence of the classical solution of a modified Stefan problem (Gibbs–Thomson law),” Matem. Sb., 183, No. 2, 77–101 (1992). · Zbl 0772.35087 |
| [13] | L. N. Slobodetskii, ”Sobolev generalized spaces and their application to boundary-value problems for partial differential equations,” Uch. Zap. Gertsen Leningrad Ped Inst., 197, 54–112 (1958). |
| [14] | S. L. Sobolev, Partial Differential Equations of Mathematical Physics, Dover, New York, 1964. · Zbl 0123.06508 |
| [15] | A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. · Zbl 0882.35004 |
| [16] | V. I. Voytitsky, ”The abstract spectral Stefan problem,” Uch. Zap. V. I. Vernadskii Tavrich. Nats. Univ. Ser. Mat. Mekh. Inform., and Kibern., 19(58), No. 2, 20–28 (2006). |
| [17] | V. I. Voytitsky, ”On the spectral problems generated by the linearized Stefan problem with the Gibbs–Thomson conditions,” Nelin. Gran. Probl., 17, 31–49 (2007). · Zbl 1164.35452 |
| [18] | V. I. Voytitsky, N. D. Kopachevsky, and P. A. Starkov, ”Auxiliary abstract boundary-value problems and problems of transmission,” in Contemporary Mathematics. Fundamental Trends [in Russian], Vol. 34, VINITI, Moscow, 2009, pp. 5–44. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
